AN INTERMEDIATE RING BETWEEN A POLYNOMIAL RING AND A POWER SERIES RING

Abstract

Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R; I)[x] the following subring of R[[x]]: [R; I][x] := { Sigma(i >= 0) ri(x)(i) epsilon R[[x]] : there exists 0 <= n epsilon Z such that r(i) epsilon I, for all i >= 71}. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor RN] is isomorphic to [R; I][x]. The results characterizing polynomial rings or power series rings with a certain ring property suggest a similar study to be carried out for the ring [R;I][x). In this paper, we characterize when the ring [R; I][x] is sernipotent, left Noetherian, left quasi-duo, principal left ideal, quasi-Baer, or left p.q.-Baer. New examples of these rings can be given by specializing to some particular ideals I, and some known results on polynomial rings and power series rings are corollaries of our formulations upon letting I = 0 or R.